LMFDB - Embedded newform 690.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,6,Mod(1,690)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("690.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	690.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$110.664835671$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} +32.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -25.0000 q^{5} -36.0000 q^{6} +32.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -100.000 q^{10} +372.000 q^{11} -144.000 q^{12} -382.000 q^{13} +128.000 q^{14} +225.000 q^{15} +256.000 q^{16} -846.000 q^{17} +324.000 q^{18} +1232.00 q^{19} -400.000 q^{20} -288.000 q^{21} +1488.00 q^{22} -529.000 q^{23} -576.000 q^{24} +625.000 q^{25} -1528.00 q^{26} -729.000 q^{27} +512.000 q^{28} -1302.00 q^{29} +900.000 q^{30} +5840.00 q^{31} +1024.00 q^{32} -3348.00 q^{33} -3384.00 q^{34} -800.000 q^{35} +1296.00 q^{36} -3862.00 q^{37} +4928.00 q^{38} +3438.00 q^{39} -1600.00 q^{40} -15774.0 q^{41} -1152.00 q^{42} +12524.0 q^{43} +5952.00 q^{44} -2025.00 q^{45} -2116.00 q^{46} +21888.0 q^{47} -2304.00 q^{48} -15783.0 q^{49} +2500.00 q^{50} +7614.00 q^{51} -6112.00 q^{52} -1734.00 q^{53} -2916.00 q^{54} -9300.00 q^{55} +2048.00 q^{56} -11088.0 q^{57} -5208.00 q^{58} +1908.00 q^{59} +3600.00 q^{60} -34678.0 q^{61} +23360.0 q^{62} +2592.00 q^{63} +4096.00 q^{64} +9550.00 q^{65} -13392.0 q^{66} +22892.0 q^{67} -13536.0 q^{68} +4761.00 q^{69} -3200.00 q^{70} +22596.0 q^{71} +5184.00 q^{72} +57602.0 q^{73} -15448.0 q^{74} -5625.00 q^{75} +19712.0 q^{76} +11904.0 q^{77} +13752.0 q^{78} -66352.0 q^{79} -6400.00 q^{80} +6561.00 q^{81} -63096.0 q^{82} +33744.0 q^{83} -4608.00 q^{84} +21150.0 q^{85} +50096.0 q^{86} +11718.0 q^{87} +23808.0 q^{88} +36654.0 q^{89} -8100.00 q^{90} -12224.0 q^{91} -8464.00 q^{92} -52560.0 q^{93} +87552.0 q^{94} -30800.0 q^{95} -9216.00 q^{96} +125174. q^{97} -63132.0 q^{98} +30132.0 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.0000	0.500000
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−36.0000	−0.408248
$7$	32.0000	0.246834	0.123417	−	0.992355i	$-0.460615\pi$
$7$	32.0000	0.246834	0.123417	+	0.992355i	$0.460615\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	−100.000	−0.316228
$11$	372.000	0.926960	0.463480	−	0.886107i	$-0.346600\pi$
$11$	372.000	0.926960	0.463480	+	0.886107i	$0.346600\pi$
$12$	−144.000	−0.288675
$13$	−382.000	−0.626910	−0.313455	−	0.949603i	$-0.601486\pi$
$13$	−382.000	−0.626910	−0.313455	+	0.949603i	$0.601486\pi$
$14$	128.000	0.174538
$15$	225.000	0.258199
$16$	256.000	0.250000
$17$	−846.000	−0.709983	−0.354992	−	0.934869i	$-0.615516\pi$
$17$	−846.000	−0.709983	−0.354992	+	0.934869i	$0.615516\pi$
$18$	324.000	0.235702
$19$	1232.00	0.782937	0.391468	−	0.920192i	$-0.371967\pi$
$19$	1232.00	0.782937	0.391468	+	0.920192i	$0.371967\pi$
$20$	−400.000	−0.223607
$21$	−288.000	−0.142510
$22$	1488.00	0.655460
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	−576.000	−0.204124
$25$	625.000	0.200000
$26$	−1528.00	−0.443292
$27$	−729.000	−0.192450
$28$	512.000	0.123417
$29$	−1302.00	−0.287486	−0.143743	−	0.989615i	$-0.545914\pi$
$29$	−1302.00	−0.287486	−0.143743	+	0.989615i	$0.545914\pi$
$30$	900.000	0.182574
$31$	5840.00	1.09146	0.545731	−	0.837960i	$-0.316252\pi$
$31$	5840.00	1.09146	0.545731	+	0.837960i	$0.316252\pi$
$32$	1024.00	0.176777
$33$	−3348.00	−0.535181
$34$	−3384.00	−0.502034
$35$	−800.000	−0.110387
$36$	1296.00	0.166667
$37$	−3862.00	−0.463776	−0.231888	−	0.972743i	$-0.574490\pi$
$37$	−3862.00	−0.463776	−0.231888	+	0.972743i	$0.574490\pi$
$38$	4928.00	0.553620
$39$	3438.00	0.361946
$40$	−1600.00	−0.158114
$41$	−15774.0	−1.46549	−0.732744	−	0.680505i	$-0.761761\pi$
$41$	−15774.0	−1.46549	−0.732744	+	0.680505i	$0.761761\pi$
$42$	−1152.00	−0.100770
$43$	12524.0	1.03293	0.516466	−	0.856308i	$-0.327247\pi$
$43$	12524.0	1.03293	0.516466	+	0.856308i	$0.327247\pi$
$44$	5952.00	0.463480
$45$	−2025.00	−0.149071
$46$	−2116.00	−0.147442
$47$	21888.0	1.44531	0.722656	−	0.691208i	$-0.242921\pi$
$47$	21888.0	1.44531	0.722656	+	0.691208i	$0.242921\pi$
$48$	−2304.00	−0.144338
$49$	−15783.0	−0.939073
$50$	2500.00	0.141421
$51$	7614.00	0.409909
$52$	−6112.00	−0.313455
$53$	−1734.00	−0.0847929	−0.0423964	−	0.999101i	$-0.513499\pi$
$53$	−1734.00	−0.0847929	−0.0423964	+	0.999101i	$0.513499\pi$
$54$	−2916.00	−0.136083
$55$	−9300.00	−0.414549
$56$	2048.00	0.0872690
$57$	−11088.0	−0.452029
$58$	−5208.00	−0.203283
$59$	1908.00	0.0713589	0.0356795	−	0.999363i	$-0.488640\pi$
$59$	1908.00	0.0713589	0.0356795	+	0.999363i	$0.488640\pi$
$60$	3600.00	0.129099
$61$	−34678.0	−1.19324	−0.596622	−	0.802522i	$-0.703491\pi$
$61$	−34678.0	−1.19324	−0.596622	+	0.802522i	$0.703491\pi$
$62$	23360.0	0.771780
$63$	2592.00	0.0822780
$64$	4096.00	0.125000
$65$	9550.00	0.280363
$66$	−13392.0	−0.378430
$67$	22892.0	0.623012	0.311506	−	0.950244i	$-0.399167\pi$
$67$	22892.0	0.623012	0.311506	+	0.950244i	$0.399167\pi$
$68$	−13536.0	−0.354992
$69$	4761.00	0.120386
$70$	−3200.00	−0.0780557
$71$	22596.0	0.531968	0.265984	−	0.963977i	$-0.414303\pi$
$71$	22596.0	0.531968	0.265984	+	0.963977i	$0.414303\pi$
$72$	5184.00	0.117851
$73$	57602.0	1.26512	0.632558	−	0.774513i	$-0.282005\pi$
$73$	57602.0	1.26512	0.632558	+	0.774513i	$0.282005\pi$
$74$	−15448.0	−0.327939
$75$	−5625.00	−0.115470
$76$	19712.0	0.391468
$77$	11904.0	0.228805
$78$	13752.0	0.255935
$79$	−66352.0	−1.19615	−0.598076	−	0.801439i	$-0.704068\pi$
$79$	−66352.0	−1.19615	−0.598076	+	0.801439i	$0.704068\pi$
$80$	−6400.00	−0.111803
$81$	6561.00	0.111111
$82$	−63096.0	−1.03626
$83$	33744.0	0.537652	0.268826	−	0.963189i	$-0.413364\pi$
$83$	33744.0	0.537652	0.268826	+	0.963189i	$0.413364\pi$
$84$	−4608.00	−0.0712548
$85$	21150.0	0.317514
$86$	50096.0	0.730394
$87$	11718.0	0.165980
$88$	23808.0	0.327730
$89$	36654.0	0.490508	0.245254	−	0.969459i	$-0.421129\pi$
$89$	36654.0	0.490508	0.245254	+	0.969459i	$0.421129\pi$
$90$	−8100.00	−0.105409
$91$	−12224.0	−0.154743
$92$	−8464.00	−0.104257
$93$	−52560.0	−0.630156
$94$	87552.0	1.02199
$95$	−30800.0	−0.350140
$96$	−9216.00	−0.102062
$97$	125174.	1.35078	0.675390	−	0.737460i	$-0.263975\pi$
$97$	125174.	1.35078	0.675390	+	0.737460i	$0.263975\pi$
$98$	−63132.0	−0.664025
$99$	30132.0	0.308987
$100$	10000.0	0.100000
$101$	86394.0	0.842714	0.421357	−	0.906895i	$-0.361554\pi$
$101$	86394.0	0.842714	0.421357	+	0.906895i	$0.361554\pi$
$102$	30456.0	0.289849
$103$	142952.	1.32769	0.663846	−	0.747870i	$-0.268923\pi$
$103$	142952.	1.32769	0.663846	+	0.747870i	$0.268923\pi$
$104$	−24448.0	−0.221646
$105$	7200.00	0.0637322
$106$	−6936.00	−0.0599576
$107$	153792.	1.29860	0.649298	−	0.760534i	$-0.275063\pi$
$107$	153792.	1.29860	0.649298	+	0.760534i	$0.275063\pi$
$108$	−11664.0	−0.0962250
$109$	27290.0	0.220007	0.110004	−	0.993931i	$-0.464914\pi$
$109$	27290.0	0.220007	0.110004	+	0.993931i	$0.464914\pi$
$110$	−37200.0	−0.293131
$111$	34758.0	0.267761
$112$	8192.00	0.0617085
$113$	140034.	1.03166	0.515831	−	0.856690i	$-0.327483\pi$
$113$	140034.	1.03166	0.515831	+	0.856690i	$0.327483\pi$
$114$	−44352.0	−0.319633
$115$	13225.0	0.0932505
$116$	−20832.0	−0.143743
$117$	−30942.0	−0.208970
$118$	7632.00	0.0504584
$119$	−27072.0	−0.175248
$120$	14400.0	0.0912871
$121$	−22667.0	−0.140744
$122$	−138712.	−0.843751
$123$	141966.	0.846100
$124$	93440.0	0.545731
$125$	−15625.0	−0.0894427
$126$	10368.0	0.0581793
$127$	109244.	0.601019	0.300510	−	0.953779i	$-0.402843\pi$
$127$	109244.	0.601019	0.300510	+	0.953779i	$0.402843\pi$
$128$	16384.0	0.0883883
$129$	−112716.	−0.596364
$130$	38200.0	0.198246
$131$	−53100.0	−0.270344	−0.135172	−	0.990822i	$-0.543159\pi$
$131$	−53100.0	−0.270344	−0.135172	+	0.990822i	$0.543159\pi$
$132$	−53568.0	−0.267590
$133$	39424.0	0.193255
$134$	91568.0	0.440536
$135$	18225.0	0.0860663
$136$	−54144.0	−0.251017
$137$	−282942.	−1.28794	−0.643971	−	0.765050i	$-0.722714\pi$
$137$	−282942.	−1.28794	−0.643971	+	0.765050i	$0.722714\pi$
$138$	19044.0	0.0851257
$139$	367628.	1.61388	0.806941	−	0.590633i	$-0.201122\pi$
$139$	367628.	1.61388	0.806941	+	0.590633i	$0.201122\pi$
$140$	−12800.0	−0.0551937
$141$	−196992.	−0.834451
$142$	90384.0	0.376158
$143$	−142104.	−0.581121
$144$	20736.0	0.0833333
$145$	32550.0	0.128567
$146$	230408.	0.894572
$147$	142047.	0.542174
$148$	−61792.0	−0.231888
$149$	11034.0	0.0407162	0.0203581	−	0.999793i	$-0.493519\pi$
$149$	11034.0	0.0407162	0.0203581	+	0.999793i	$0.493519\pi$
$150$	−22500.0	−0.0816497
$151$	452768.	1.61597	0.807985	−	0.589203i	$-0.200558\pi$
$151$	452768.	1.61597	0.807985	+	0.589203i	$0.200558\pi$
$152$	78848.0	0.276810
$153$	−68526.0	−0.236661
$154$	47616.0	0.161790
$155$	−146000.	−0.488117
$156$	55008.0	0.180973
$157$	371474.	1.20276	0.601380	−	0.798963i	$-0.294618\pi$
$157$	371474.	1.20276	0.601380	+	0.798963i	$0.294618\pi$
$158$	−265408.	−0.845807
$159$	15606.0	0.0489552
$160$	−25600.0	−0.0790569
$161$	−16928.0	−0.0514684
$162$	26244.0	0.0785674
$163$	−357676.	−1.05444	−0.527219	−	0.849730i	$-0.676765\pi$
$163$	−357676.	−1.05444	−0.527219	+	0.849730i	$0.676765\pi$
$164$	−252384.	−0.732744
$165$	83700.0	0.239340
$166$	134976.	0.380177
$167$	533808.	1.48113	0.740566	−	0.671983i	$-0.234557\pi$
$167$	533808.	1.48113	0.740566	+	0.671983i	$0.234557\pi$
$168$	−18432.0	−0.0503848
$169$	−225369.	−0.606984
$170$	84600.0	0.224516
$171$	99792.0	0.260979
$172$	200384.	0.516466
$173$	523590.	1.33007	0.665037	−	0.746810i	$-0.268416\pi$
$173$	523590.	1.33007	0.665037	+	0.746810i	$0.268416\pi$
$174$	46872.0	0.117365
$175$	20000.0	0.0493668
$176$	95232.0	0.231740
$177$	−17172.0	−0.0411991
$178$	146616.	0.346842
$179$	−447180.	−1.04316	−0.521579	−	0.853203i	$-0.674657\pi$
$179$	−447180.	−1.04316	−0.521579	+	0.853203i	$0.674657\pi$
$180$	−32400.0	−0.0745356
$181$	160754.	0.364725	0.182362	−	0.983231i	$-0.441626\pi$
$181$	160754.	0.364725	0.182362	+	0.983231i	$0.441626\pi$
$182$	−48896.0	−0.109420
$183$	312102.	0.688920
$184$	−33856.0	−0.0737210
$185$	96550.0	0.207407
$186$	−210240.	−0.445587
$187$	−314712.	−0.658126
$188$	350208.	0.722656
$189$	−23328.0	−0.0475032
$190$	−123200.	−0.247586
$191$	659400.	1.30787	0.653936	−	0.756550i	$-0.273116\pi$
$191$	659400.	1.30787	0.653936	+	0.756550i	$0.273116\pi$
$192$	−36864.0	−0.0721688
$193$	−656494.	−1.26864	−0.634319	−	0.773072i	$-0.718719\pi$
$193$	−656494.	−1.26864	−0.634319	+	0.773072i	$0.718719\pi$
$194$	500696.	0.955146
$195$	−85950.0	−0.161867
$196$	−252528.	−0.469537
$197$	−852738.	−1.56549	−0.782745	−	0.622343i	$-0.786181\pi$
$197$	−852738.	−1.56549	−0.782745	+	0.622343i	$0.786181\pi$
$198$	120528.	0.218487
$199$	412496.	0.738392	0.369196	−	0.929352i	$-0.379633\pi$
$199$	412496.	0.738392	0.369196	+	0.929352i	$0.379633\pi$
$200$	40000.0	0.0707107
$201$	−206028.	−0.359696
$202$	345576.	0.595889
$203$	−41664.0	−0.0709612
$204$	121824.	0.204954
$205$	394350.	0.655386
$206$	571808.	0.938820
$207$	−42849.0	−0.0695048
$208$	−97792.0	−0.156727
$209$	458304.	0.725751
$210$	28800.0	0.0450655
$211$	1.01373e6	1.56753	0.783767	−	0.621055i	$-0.213296\pi$
$211$	1.01373e6	1.56753	0.783767	+	0.621055i	$0.213296\pi$
$212$	−27744.0	−0.0423964
$213$	−203364.	−0.307132
$214$	615168.	0.918247
$215$	−313100.	−0.461941
$216$	−46656.0	−0.0680414
$217$	186880.	0.269410
$218$	109160.	0.155569
$219$	−518418.	−0.730415
$220$	−148800.	−0.207275
$221$	323172.	0.445095
$222$	139032.	0.189336
$223$	−182164.	−0.245302	−0.122651	−	0.992450i	$-0.539140\pi$
$223$	−182164.	−0.245302	−0.122651	+	0.992450i	$0.539140\pi$
$224$	32768.0	0.0436345
$225$	50625.0	0.0666667
$226$	560136.	0.729495
$227$	−70848.0	−0.0912563	−0.0456282	−	0.998958i	$-0.514529\pi$
$227$	−70848.0	−0.0912563	−0.0456282	+	0.998958i	$0.514529\pi$
$228$	−177408.	−0.226014
$229$	−116254.	−0.146494	−0.0732469	−	0.997314i	$-0.523336\pi$
$229$	−116254.	−0.146494	−0.0732469	+	0.997314i	$0.523336\pi$
$230$	52900.0	0.0659380
$231$	−107136.	−0.132101
$232$	−83328.0	−0.101641
$233$	1.30915e6	1.57979	0.789893	−	0.613245i	$-0.210136\pi$
$233$	1.30915e6	1.57979	0.789893	+	0.613245i	$0.210136\pi$
$234$	−123768.	−0.147764
$235$	−547200.	−0.646363
$236$	30528.0	0.0356795
$237$	597168.	0.690599
$238$	−108288.	−0.123919
$239$	−836220.	−0.946947	−0.473474	−	0.880808i	$-0.657000\pi$
$239$	−836220.	−0.946947	−0.473474	+	0.880808i	$0.657000\pi$
$240$	57600.0	0.0645497
$241$	640490.	0.710346	0.355173	−	0.934801i	$-0.384422\pi$
$241$	640490.	0.710346	0.355173	+	0.934801i	$0.384422\pi$
$242$	−90668.0	−0.0995212
$243$	−59049.0	−0.0641500
$244$	−554848.	−0.596622
$245$	394575.	0.419966
$246$	567864.	0.598283
$247$	−470624.	−0.490831
$248$	373760.	0.385890
$249$	−303696.	−0.310414
$250$	−62500.0	−0.0632456
$251$	1.45040e6	1.45313	0.726565	−	0.687097i	$-0.241115\pi$
$251$	1.45040e6	1.45313	0.726565	+	0.687097i	$0.241115\pi$
$252$	41472.0	0.0411390
$253$	−196788.	−0.193285
$254$	436976.	0.424985
$255$	−190350.	−0.183317
$256$	65536.0	0.0625000
$257$	721218.	0.681136	0.340568	−	0.940220i	$-0.389381\pi$
$257$	721218.	0.681136	0.340568	+	0.940220i	$0.389381\pi$
$258$	−450864.	−0.421693
$259$	−123584.	−0.114476
$260$	152800.	0.140181
$261$	−105462.	−0.0958285
$262$	−212400.	−0.191162
$263$	762288.	0.679563	0.339781	−	0.940504i	$-0.389647\pi$
$263$	762288.	0.679563	0.339781	+	0.940504i	$0.389647\pi$
$264$	−214272.	−0.189215
$265$	43350.0	0.0379205
$266$	157696.	0.136652
$267$	−329886.	−0.283195
$268$	366272.	0.311506
$269$	−1.61187e6	−1.35815	−0.679077	−	0.734067i	$-0.737620\pi$
$269$	−1.61187e6	−1.35815	−0.679077	+	0.734067i	$0.737620\pi$
$270$	72900.0	0.0608581
$271$	−817048.	−0.675810	−0.337905	−	0.941180i	$-0.609718\pi$
$271$	−817048.	−0.675810	−0.337905	+	0.941180i	$0.609718\pi$
$272$	−216576.	−0.177496
$273$	110016.	0.0893407
$274$	−1.13177e6	−0.910712
$275$	232500.	0.185392
$276$	76176.0	0.0601929
$277$	1.58794e6	1.24347	0.621733	−	0.783229i	$-0.286429\pi$
$277$	1.58794e6	1.24347	0.621733	+	0.783229i	$0.286429\pi$
$278$	1.47051e6	1.14119
$279$	473040.	0.363821
$280$	−51200.0	−0.0390279
$281$	−1.03365e6	−0.780922	−0.390461	−	0.920619i	$-0.627684\pi$
$281$	−1.03365e6	−0.780922	−0.390461	+	0.920619i	$0.627684\pi$
$282$	−787968.	−0.590046
$283$	−560620.	−0.416105	−0.208052	−	0.978118i	$-0.566712\pi$
$283$	−560620.	−0.416105	−0.208052	+	0.978118i	$0.566712\pi$
$284$	361536.	0.265984
$285$	277200.	0.202153
$286$	−568416.	−0.410914
$287$	−504768.	−0.361732
$288$	82944.0	0.0589256
$289$	−704141.	−0.495924
$290$	130200.	0.0909109
$291$	−1.12657e6	−0.779874
$292$	921632.	0.632558
$293$	−2.41287e6	−1.64197	−0.820984	−	0.570951i	$-0.806575\pi$
$293$	−2.41287e6	−1.64197	−0.820984	+	0.570951i	$0.806575\pi$
$294$	568188.	0.383375
$295$	−47700.0	−0.0319127
$296$	−247168.	−0.163969
$297$	−271188.	−0.178394
$298$	44136.0	0.0287907
$299$	202078.	0.130720
$300$	−90000.0	−0.0577350
$301$	400768.	0.254963
$302$	1.81107e6	1.14266
$303$	−777546.	−0.486541
$304$	315392.	0.195734
$305$	866950.	0.533635
$306$	−274104.	−0.167345
$307$	2.25729e6	1.36692	0.683458	−	0.729990i	$-0.260475\pi$
$307$	2.25729e6	1.36692	0.683458	+	0.729990i	$0.260475\pi$
$308$	190464.	0.114403
$309$	−1.28657e6	−0.766543
$310$	−584000.	−0.345151
$311$	−2.02948e6	−1.18982	−0.594912	−	0.803791i	$-0.702813\pi$
$311$	−2.02948e6	−1.18982	−0.594912	+	0.803791i	$0.702813\pi$
$312$	220032.	0.127967
$313$	2.48625e6	1.43444	0.717221	−	0.696845i	$-0.245414\pi$
$313$	2.48625e6	1.43444	0.717221	+	0.696845i	$0.245414\pi$
$314$	1.48590e6	0.850480
$315$	−64800.0	−0.0367958
$316$	−1.06163e6	−0.598076
$317$	2.10065e6	1.17410	0.587052	−	0.809549i	$-0.300289\pi$
$317$	2.10065e6	1.17410	0.587052	+	0.809549i	$0.300289\pi$
$318$	62424.0	0.0346166
$319$	−484344.	−0.266488
$320$	−102400.	−0.0559017
$321$	−1.38413e6	−0.749745
$322$	−67712.0	−0.0363937
$323$	−1.04227e6	−0.555872
$324$	104976.	0.0555556
$325$	−238750.	−0.125382
$326$	−1.43070e6	−0.745600
$327$	−245610.	−0.127021
$328$	−1.00954e6	−0.518128
$329$	700416.	0.356752
$330$	334800.	0.169239
$331$	−2.35508e6	−1.18150	−0.590752	−	0.806853i	$-0.701169\pi$
$331$	−2.35508e6	−1.18150	−0.590752	+	0.806853i	$0.701169\pi$
$332$	539904.	0.268826
$333$	−312822.	−0.154592
$334$	2.13523e6	1.04732
$335$	−572300.	−0.278620
$336$	−73728.0	−0.0356274
$337$	−2.32191e6	−1.11371	−0.556854	−	0.830610i	$-0.687992\pi$
$337$	−2.32191e6	−1.11371	−0.556854	+	0.830610i	$0.687992\pi$
$338$	−901476.	−0.429203
$339$	−1.26031e6	−0.595630
$340$	338400.	0.158757
$341$	2.17248e6	1.01174
$342$	399168.	0.184540
$343$	−1.04288e6	−0.478629
$344$	801536.	0.365197
$345$	−119025.	−0.0538382
$346$	2.09436e6	0.940505
$347$	650940.	0.290213	0.145107	−	0.989416i	$-0.453647\pi$
$347$	650940.	0.290213	0.145107	+	0.989416i	$0.453647\pi$
$348$	187488.	0.0829899
$349$	373310.	0.164061	0.0820306	−	0.996630i	$-0.473859\pi$
$349$	373310.	0.164061	0.0820306	+	0.996630i	$0.473859\pi$
$350$	80000.0	0.0349076
$351$	278478.	0.120649
$352$	380928.	0.163865
$353$	−2.92097e6	−1.24764	−0.623822	−	0.781566i	$-0.714421\pi$
$353$	−2.92097e6	−1.24764	−0.623822	+	0.781566i	$0.714421\pi$
$354$	−68688.0	−0.0291322
$355$	−564900.	−0.237903
$356$	586464.	0.245254
$357$	243648.	0.101179
$358$	−1.78872e6	−0.737624
$359$	−3.58286e6	−1.46722	−0.733608	−	0.679573i	$-0.762165\pi$
$359$	−3.58286e6	−1.46722	−0.733608	+	0.679573i	$0.762165\pi$
$360$	−129600.	−0.0527046
$361$	−958275.	−0.387010
$362$	643016.	0.257899
$363$	204003.	0.0812587
$364$	−195584.	−0.0773713
$365$	−1.44005e6	−0.565777
$366$	1.24841e6	0.487140
$367$	2.85913e6	1.10807	0.554037	−	0.832492i	$-0.313087\pi$
$367$	2.85913e6	1.10807	0.554037	+	0.832492i	$0.313087\pi$
$368$	−135424.	−0.0521286
$369$	−1.27769e6	−0.488496
$370$	386200.	0.146659
$371$	−55488.0	−0.0209298
$372$	−840960.	−0.315078
$373$	−1.57197e6	−0.585023	−0.292512	−	0.956262i	$-0.594491\pi$
$373$	−1.57197e6	−0.585023	−0.292512	+	0.956262i	$0.594491\pi$
$374$	−1.25885e6	−0.465366
$375$	140625.	0.0516398
$376$	1.40083e6	0.510995
$377$	497364.	0.180227
$378$	−93312.0	−0.0335898
$379$	492032.	0.175952	0.0879762	−	0.996123i	$-0.471960\pi$
$379$	492032.	0.175952	0.0879762	+	0.996123i	$0.471960\pi$
$380$	−492800.	−0.175070
$381$	−983196.	−0.346999
$382$	2.63760e6	0.924806
$383$	−1.82758e6	−0.636617	−0.318309	−	0.947987i	$-0.603115\pi$
$383$	−1.82758e6	−0.636617	−0.318309	+	0.947987i	$0.603115\pi$
$384$	−147456.	−0.0510310
$385$	−297600.	−0.102325
$386$	−2.62598e6	−0.897062
$387$	1.01444e6	0.344311
$388$	2.00278e6	0.675390
$389$	−3.94810e6	−1.32286	−0.661431	−	0.750006i	$-0.730050\pi$
$389$	−3.94810e6	−1.32286	−0.661431	+	0.750006i	$0.730050\pi$
$390$	−343800.	−0.114458
$391$	447534.	0.148042
$392$	−1.01011e6	−0.332012
$393$	477900.	0.156083
$394$	−3.41095e6	−1.10697
$395$	1.65880e6	0.534935
$396$	482112.	0.154493
$397$	−745630.	−0.237436	−0.118718	−	0.992928i	$-0.537878\pi$
$397$	−745630.	−0.237436	−0.118718	+	0.992928i	$0.537878\pi$
$398$	1.64998e6	0.522122
$399$	−354816.	−0.111576
$400$	160000.	0.0500000
$401$	5.08738e6	1.57991	0.789957	−	0.613162i	$-0.210103\pi$
$401$	5.08738e6	1.57991	0.789957	+	0.613162i	$0.210103\pi$
$402$	−824112.	−0.254344
$403$	−2.23088e6	−0.684248
$404$	1.38230e6	0.421357
$405$	−164025.	−0.0496904
$406$	−166656.	−0.0501771
$407$	−1.43666e6	−0.429902
$408$	487296.	0.144925
$409$	2.12844e6	0.629149	0.314575	−	0.949233i	$-0.398138\pi$
$409$	2.12844e6	0.629149	0.314575	+	0.949233i	$0.398138\pi$
$410$	1.57740e6	0.463428
$411$	2.54648e6	0.743593
$412$	2.28723e6	0.663846
$413$	61056.0	0.0176138
$414$	−171396.	−0.0491473
$415$	−843600.	−0.240445
$416$	−391168.	−0.110823
$417$	−3.30865e6	−0.931775
$418$	1.83322e6	0.513184
$419$	5.44016e6	1.51383	0.756915	−	0.653514i	$-0.226706\pi$
$419$	5.44016e6	1.51383	0.756915	+	0.653514i	$0.226706\pi$
$420$	115200.	0.0318661
$421$	−1.23103e6	−0.338504	−0.169252	−	0.985573i	$-0.554135\pi$
$421$	−1.23103e6	−0.338504	−0.169252	+	0.985573i	$0.554135\pi$
$422$	4.05493e6	1.10841
$423$	1.77293e6	0.481770
$424$	−110976.	−0.0299788
$425$	−528750.	−0.141997
$426$	−813456.	−0.217175
$427$	−1.10970e6	−0.294533
$428$	2.46067e6	0.649298
$429$	1.27894e6	0.335510
$430$	−1.25240e6	−0.326642
$431$	298056.	0.0772867	0.0386433	−	0.999253i	$-0.487696\pi$
$431$	298056.	0.0772867	0.0386433	+	0.999253i	$0.487696\pi$
$432$	−186624.	−0.0481125
$433$	569126.	0.145878	0.0729388	−	0.997336i	$-0.476762\pi$
$433$	569126.	0.145878	0.0729388	+	0.997336i	$0.476762\pi$
$434$	747520.	0.190502
$435$	−292950.	−0.0742284
$436$	436640.	0.110004
$437$	−651728.	−0.163254
$438$	−2.07367e6	−0.516482
$439$	−4.10634e6	−1.01694	−0.508468	−	0.861081i	$-0.669788\pi$
$439$	−4.10634e6	−1.01694	−0.508468	+	0.861081i	$0.669788\pi$
$440$	−595200.	−0.146565
$441$	−1.27842e6	−0.313024
$442$	1.29269e6	0.314730
$443$	−3.03196e6	−0.734030	−0.367015	−	0.930215i	$-0.619620\pi$
$443$	−3.03196e6	−0.734030	−0.367015	+	0.930215i	$0.619620\pi$
$444$	556128.	0.133881
$445$	−916350.	−0.219362
$446$	−728656.	−0.173454
$447$	−99306.0	−0.0235075
$448$	131072.	0.0308542
$449$	1.46039e6	0.341865	0.170932	−	0.985283i	$-0.445322\pi$
$449$	1.46039e6	0.341865	0.170932	+	0.985283i	$0.445322\pi$
$450$	202500.	0.0471405
$451$	−5.86793e6	−1.35845
$452$	2.24054e6	0.515831
$453$	−4.07491e6	−0.932981
$454$	−283392.	−0.0645280
$455$	305600.	0.0692030
$456$	−709632.	−0.159816
$457$	7.47088e6	1.67333	0.836664	−	0.547716i	$-0.184503\pi$
$457$	7.47088e6	1.67333	0.836664	+	0.547716i	$0.184503\pi$
$458$	−465016.	−0.103587
$459$	616734.	0.136636
$460$	211600.	0.0466252
$461$	−1.33997e6	−0.293660	−0.146830	−	0.989162i	$-0.546907\pi$
$461$	−1.33997e6	−0.293660	−0.146830	+	0.989162i	$0.546907\pi$
$462$	−428544.	−0.0934094
$463$	3.31408e6	0.718474	0.359237	−	0.933246i	$-0.383037\pi$
$463$	3.31408e6	0.718474	0.359237	+	0.933246i	$0.383037\pi$
$464$	−333312.	−0.0718714
$465$	1.31400e6	0.281814
$466$	5.23658e6	1.11708
$467$	2.41502e6	0.512424	0.256212	−	0.966621i	$-0.417526\pi$
$467$	2.41502e6	0.512424	0.256212	+	0.966621i	$0.417526\pi$
$468$	−495072.	−0.104485
$469$	732544.	0.153781
$470$	−2.18880e6	−0.457048
$471$	−3.34327e6	−0.694414
$472$	122112.	0.0252292
$473$	4.65893e6	0.957488
$474$	2.38867e6	0.488327
$475$	770000.	0.156587
$476$	−433152.	−0.0876240
$477$	−140454.	−0.0282643
$478$	−3.34488e6	−0.669593
$479$	2.13461e6	0.425088	0.212544	−	0.977151i	$-0.431825\pi$
$479$	2.13461e6	0.425088	0.212544	+	0.977151i	$0.431825\pi$
$480$	230400.	0.0456435
$481$	1.47528e6	0.290745
$482$	2.56196e6	0.502290
$483$	152352.	0.0297153
$484$	−362672.	−0.0703721
$485$	−3.12935e6	−0.604087
$486$	−236196.	−0.0453609
$487$	2.48361e6	0.474527	0.237264	−	0.971445i	$-0.423749\pi$
$487$	2.48361e6	0.474527	0.237264	+	0.971445i	$0.423749\pi$
$488$	−2.21939e6	−0.421876
$489$	3.21908e6	0.608780
$490$	1.57830e6	0.296961
$491$	2.81221e6	0.526434	0.263217	−	0.964737i	$-0.415216\pi$
$491$	2.81221e6	0.526434	0.263217	+	0.964737i	$0.415216\pi$
$492$	2.27146e6	0.423050
$493$	1.10149e6	0.204110
$494$	−1.88250e6	−0.347070
$495$	−753300.	−0.138183
$496$	1.49504e6	0.272865
$497$	723072.	0.131308
$498$	−1.21478e6	−0.219496
$499$	7.59100e6	1.36473	0.682367	−	0.731010i	$-0.260951\pi$
$499$	7.59100e6	1.36473	0.682367	+	0.731010i	$0.260951\pi$
$500$	−250000.	−0.0447214
$501$	−4.80427e6	−0.855132
$502$	5.80162e6	1.02752
$503$	−4.79856e6	−0.845651	−0.422825	−	0.906211i	$-0.638962\pi$
$503$	−4.79856e6	−0.845651	−0.422825	+	0.906211i	$0.638962\pi$
$504$	165888.	0.0290897
$505$	−2.15985e6	−0.376873
$506$	−787152.	−0.136673
$507$	2.02832e6	0.350443
$508$	1.74790e6	0.300510
$509$	−2.45689e6	−0.420330	−0.210165	−	0.977666i	$-0.567400\pi$
$509$	−2.45689e6	−0.420330	−0.210165	+	0.977666i	$0.567400\pi$
$510$	−761400.	−0.129625
$511$	1.84326e6	0.312274
$512$	262144.	0.0441942
$513$	−898128.	−0.150676
$514$	2.88487e6	0.481636
$515$	−3.57380e6	−0.593762
$516$	−1.80346e6	−0.298182
$517$	8.14234e6	1.33975
$518$	−494336.	−0.0809465
$519$	−4.71231e6	−0.767919
$520$	611200.	0.0991231
$521$	−1.23443e7	−1.99239	−0.996194	−	0.0871674i	$-0.972219\pi$
$521$	−1.23443e7	−1.99239	−0.996194	+	0.0871674i	$0.972219\pi$
$522$	−421848.	−0.0677610
$523$	−3.78572e6	−0.605193	−0.302596	−	0.953119i	$-0.597853\pi$
$523$	−3.78572e6	−0.605193	−0.302596	+	0.953119i	$0.597853\pi$
$524$	−849600.	−0.135172
$525$	−180000.	−0.0285019
$526$	3.04915e6	0.480524
$527$	−4.94064e6	−0.774920
$528$	−857088.	−0.133795
$529$	279841.	0.0434783
$530$	173400.	0.0268139
$531$	154548.	0.0237863
$532$	630784.	0.0966277
$533$	6.02567e6	0.918728
$534$	−1.31954e6	−0.200249
$535$	−3.84480e6	−0.580750
$536$	1.46509e6	0.220268
$537$	4.02462e6	0.602267
$538$	−6.44748e6	−0.960361
$539$	−5.87128e6	−0.870484
$540$	291600.	0.0430331
$541$	−4.32426e6	−0.635212	−0.317606	−	0.948223i	$-0.602879\pi$
$541$	−4.32426e6	−0.635212	−0.317606	+	0.948223i	$0.602879\pi$
$542$	−3.26819e6	−0.477870
$543$	−1.44679e6	−0.210574
$544$	−866304.	−0.125508
$545$	−682250.	−0.0983903
$546$	440064.	0.0631734
$547$	−1.10591e7	−1.58035	−0.790174	−	0.612883i	$-0.790010\pi$
$547$	−1.10591e7	−1.58035	−0.790174	+	0.612883i	$0.790010\pi$
$548$	−4.52707e6	−0.643971
$549$	−2.80892e6	−0.397748
$550$	930000.	0.131092
$551$	−1.60406e6	−0.225083
$552$	304704.	0.0425628
$553$	−2.12326e6	−0.295251
$554$	6.35175e6	0.879264
$555$	−868950.	−0.119746
$556$	5.88205e6	0.806941
$557$	−1.10752e7	−1.51257	−0.756284	−	0.654243i	$-0.772987\pi$
$557$	−1.10752e7	−1.51257	−0.756284	+	0.654243i	$0.772987\pi$
$558$	1.89216e6	0.257260
$559$	−4.78417e6	−0.647555
$560$	−204800.	−0.0275969
$561$	2.83241e6	0.379969
$562$	−4.13460e6	−0.552195
$563$	−3.37164e6	−0.448302	−0.224151	−	0.974554i	$-0.571961\pi$
$563$	−3.37164e6	−0.448302	−0.224151	+	0.974554i	$0.571961\pi$
$564$	−3.15187e6	−0.417225
$565$	−3.50085e6	−0.461373
$566$	−2.24248e6	−0.294230
$567$	209952.	0.0274260
$568$	1.44614e6	0.188079
$569$	−781410.	−0.101181	−0.0505904	−	0.998719i	$-0.516110\pi$
$569$	−781410.	−0.101181	−0.0505904	+	0.998719i	$0.516110\pi$
$570$	1.10880e6	0.142944
$571$	2.58882e6	0.332285	0.166143	−	0.986102i	$-0.446869\pi$
$571$	2.58882e6	0.332285	0.166143	+	0.986102i	$0.446869\pi$
$572$	−2.27366e6	−0.290560
$573$	−5.93460e6	−0.755101
$574$	−2.01907e6	−0.255783
$575$	−330625.	−0.0417029
$576$	331776.	0.0416667
$577$	−2.28108e6	−0.285234	−0.142617	−	0.989778i	$-0.545552\pi$
$577$	−2.28108e6	−0.285234	−0.142617	+	0.989778i	$0.545552\pi$
$578$	−2.81656e6	−0.350671
$579$	5.90845e6	0.732448
$580$	520800.	0.0642837
$581$	1.07981e6	0.132711
$582$	−4.50626e6	−0.551454
$583$	−645048.	−0.0785997
$584$	3.68653e6	0.447286
$585$	773550.	0.0934542
$586$	−9.65148e6	−1.16105
$587$	577956.	0.0692308	0.0346154	−	0.999401i	$-0.488979\pi$
$587$	577956.	0.0692308	0.0346154	+	0.999401i	$0.488979\pi$
$588$	2.27275e6	0.271087
$589$	7.19488e6	0.854546
$590$	−190800.	−0.0225657
$591$	7.67464e6	0.903836
$592$	−988672.	−0.115944
$593$	−9.41105e6	−1.09901	−0.549505	−	0.835491i	$-0.685184\pi$
$593$	−9.41105e6	−1.09901	−0.549505	+	0.835491i	$0.685184\pi$
$594$	−1.08475e6	−0.126143
$595$	676800.	0.0783733
$596$	176544.	0.0203581
$597$	−3.71246e6	−0.426311
$598$	808312.	0.0924328
$599$	−9.98084e6	−1.13658	−0.568290	−	0.822828i	$-0.692395\pi$
$599$	−9.98084e6	−1.13658	−0.568290	+	0.822828i	$0.692395\pi$
$600$	−360000.	−0.0408248
$601$	−2.09349e6	−0.236421	−0.118210	−	0.992989i	$-0.537716\pi$
$601$	−2.09349e6	−0.236421	−0.118210	+	0.992989i	$0.537716\pi$
$602$	1.60307e6	0.180286
$603$	1.85425e6	0.207671
$604$	7.24429e6	0.807985
$605$	566675.	0.0629427
$606$	−3.11018e6	−0.344037
$607$	7.16035e6	0.788792	0.394396	−	0.918941i	$-0.370954\pi$
$607$	7.16035e6	0.788792	0.394396	+	0.918941i	$0.370954\pi$
$608$	1.26157e6	0.138405
$609$	374976.	0.0409695
$610$	3.46780e6	0.377337
$611$	−8.36122e6	−0.906080
$612$	−1.09642e6	−0.118331
$613$	1.32798e7	1.42739	0.713694	−	0.700458i	$-0.247021\pi$
$613$	1.32798e7	1.42739	0.713694	+	0.700458i	$0.247021\pi$
$614$	9.02917e6	0.966556
$615$	−3.54915e6	−0.378387
$616$	761856.	0.0808949
$617$	1.15423e6	0.122061	0.0610306	−	0.998136i	$-0.480561\pi$
$617$	1.15423e6	0.122061	0.0610306	+	0.998136i	$0.480561\pi$
$618$	−5.14627e6	−0.542028
$619$	−1.62292e7	−1.70243	−0.851217	−	0.524813i	$-0.824135\pi$
$619$	−1.62292e7	−1.70243	−0.851217	+	0.524813i	$0.824135\pi$
$620$	−2.33600e6	−0.244058
$621$	385641.	0.0401286
$622$	−8.11790e6	−0.841333
$623$	1.17293e6	0.121074
$624$	880128.	0.0904866
$625$	390625.	0.0400000
$626$	9.94498e6	1.01430
$627$	−4.12474e6	−0.419013
$628$	5.94358e6	0.601380
$629$	3.26725e6	0.329273
$630$	−259200.	−0.0260186
$631$	−3.74248e6	−0.374185	−0.187092	−	0.982342i	$-0.559906\pi$
$631$	−3.74248e6	−0.374185	−0.187092	+	0.982342i	$0.559906\pi$
$632$	−4.24653e6	−0.422904
$633$	−9.12359e6	−0.905016
$634$	8.40262e6	0.830217
$635$	−2.73110e6	−0.268784
$636$	249696.	0.0244776
$637$	6.02911e6	0.588714
$638$	−1.93738e6	−0.188435
$639$	1.83028e6	0.177323
$640$	−409600.	−0.0395285
$641$	−4.58167e6	−0.440431	−0.220216	−	0.975451i	$-0.570676\pi$
$641$	−4.58167e6	−0.440431	−0.220216	+	0.975451i	$0.570676\pi$
$642$	−5.53651e6	−0.530150
$643$	5.02392e6	0.479198	0.239599	−	0.970872i	$-0.422984\pi$
$643$	5.02392e6	0.479198	0.239599	+	0.970872i	$0.422984\pi$
$644$	−270848.	−0.0257342
$645$	2.81790e6	0.266702
$646$	−4.16909e6	−0.393061
$647$	2.10269e6	0.197476	0.0987380	−	0.995113i	$-0.468519\pi$
$647$	2.10269e6	0.197476	0.0987380	+	0.995113i	$0.468519\pi$
$648$	419904.	0.0392837
$649$	709776.	0.0661469
$650$	−955000.	−0.0886584
$651$	−1.68192e6	−0.155544
$652$	−5.72282e6	−0.527219
$653$	−2.09388e7	−1.92163	−0.960815	−	0.277191i	$-0.910597\pi$
$653$	−2.09388e7	−1.92163	−0.960815	+	0.277191i	$0.910597\pi$
$654$	−982440.	−0.0898176
$655$	1.32750e6	0.120901
$656$	−4.03814e6	−0.366372
$657$	4.66576e6	0.421705
$658$	2.80166e6	0.252262
$659$	−6.17771e6	−0.554133	−0.277066	−	0.960851i	$-0.589362\pi$
$659$	−6.17771e6	−0.554133	−0.277066	+	0.960851i	$0.589362\pi$
$660$	1.33920e6	0.119670
$661$	1.05622e7	0.940267	0.470133	−	0.882595i	$-0.344206\pi$
$661$	1.05622e7	0.940267	0.470133	+	0.882595i	$0.344206\pi$
$662$	−9.42030e6	−0.835449
$663$	−2.90855e6	−0.256976
$664$	2.15962e6	0.190089
$665$	−985600.	−0.0864264
$666$	−1.25129e6	−0.109313
$667$	688758.	0.0599449
$668$	8.54093e6	0.740566
$669$	1.63948e6	0.141625
$670$	−2.28920e6	−0.197014
$671$	−1.29002e7	−1.10609
$672$	−294912.	−0.0251924
$673$	3.19500e6	0.271915	0.135958	−	0.990715i	$-0.456589\pi$
$673$	3.19500e6	0.271915	0.135958	+	0.990715i	$0.456589\pi$
$674$	−9.28766e6	−0.787511
$675$	−455625.	−0.0384900
$676$	−3.60590e6	−0.303492
$677$	−7.38447e6	−0.619224	−0.309612	−	0.950863i	$-0.600199\pi$
$677$	−7.38447e6	−0.619224	−0.309612	+	0.950863i	$0.600199\pi$
$678$	−5.04122e6	−0.421174
$679$	4.00557e6	0.333418
$680$	1.35360e6	0.112258
$681$	637632.	0.0526869
$682$	8.68992e6	0.715410
$683$	−2.00003e7	−1.64053	−0.820265	−	0.571984i	$-0.806174\pi$
$683$	−2.00003e7	−1.64053	−0.820265	+	0.571984i	$0.806174\pi$
$684$	1.59667e6	0.130489
$685$	7.07355e6	0.575985
$686$	−4.17152e6	−0.338442
$687$	1.04629e6	0.0845782
$688$	3.20614e6	0.258233
$689$	662388.	0.0531575
$690$	−476100.	−0.0380693
$691$	1.97883e7	1.57657	0.788286	−	0.615308i	$-0.210969\pi$
$691$	1.97883e7	1.57657	0.788286	+	0.615308i	$0.210969\pi$
$692$	8.37744e6	0.665037
$693$	964224.	0.0762684
$694$	2.60376e6	0.205212
$695$	−9.19070e6	−0.721750
$696$	749952.	0.0586827
$697$	1.33448e7	1.04047
$698$	1.49324e6	0.116009
$699$	−1.17823e7	−0.912090
$700$	320000.	0.0246834
$701$	−1.57309e6	−0.120909	−0.0604543	−	0.998171i	$-0.519255\pi$
$701$	−1.57309e6	−0.120909	−0.0604543	+	0.998171i	$0.519255\pi$
$702$	1.11391e6	0.0853116
$703$	−4.75798e6	−0.363107
$704$	1.52371e6	0.115870
$705$	4.92480e6	0.373178
$706$	−1.16839e7	−0.882218
$707$	2.76461e6	0.208010
$708$	−274752.	−0.0205995
$709$	−1.21073e7	−0.904545	−0.452272	−	0.891880i	$-0.649387\pi$
$709$	−1.21073e7	−0.904545	−0.452272	+	0.891880i	$0.649387\pi$
$710$	−2.25960e6	−0.168223
$711$	−5.37451e6	−0.398717
$712$	2.34586e6	0.173421
$713$	−3.08936e6	−0.227586
$714$	974592.	0.0715447
$715$	3.55260e6	0.259885
$716$	−7.15488e6	−0.521579
$717$	7.52598e6	0.546720
$718$	−1.43315e7	−1.03748
$719$	1.38124e7	0.996430	0.498215	−	0.867054i	$-0.333989\pi$
$719$	1.38124e7	0.996430	0.498215	+	0.867054i	$0.333989\pi$
$720$	−518400.	−0.0372678
$721$	4.57446e6	0.327719
$722$	−3.83310e6	−0.273657
$723$	−5.76441e6	−0.410118
$724$	2.57206e6	0.182362
$725$	−813750.	−0.0574971
$726$	816012.	0.0574586
$727$	−455224.	−0.0319440	−0.0159720	−	0.999872i	$-0.505084\pi$
$727$	−455224.	−0.0319440	−0.0159720	+	0.999872i	$0.505084\pi$
$728$	−782336.	−0.0547098
$729$	531441.	0.0370370
$730$	−5.76020e6	−0.400065
$731$	−1.05953e7	−0.733365
$732$	4.99363e6	0.344460
$733$	−1.30986e7	−0.900459	−0.450230	−	0.892913i	$-0.648658\pi$
$733$	−1.30986e7	−0.900459	−0.450230	+	0.892913i	$0.648658\pi$
$734$	1.14365e7	0.783526
$735$	−3.55118e6	−0.242468
$736$	−541696.	−0.0368605
$737$	8.51582e6	0.577508
$738$	−5.11078e6	−0.345419
$739$	1.87971e7	1.26613	0.633066	−	0.774098i	$-0.281796\pi$
$739$	1.87971e7	1.26613	0.633066	+	0.774098i	$0.281796\pi$
$740$	1.54480e6	0.103703
$741$	4.23562e6	0.283381
$742$	−221952.	−0.0147996
$743$	−2.06305e7	−1.37100	−0.685500	−	0.728072i	$-0.740416\pi$
$743$	−2.06305e7	−1.37100	−0.685500	+	0.728072i	$0.740416\pi$
$744$	−3.36384e6	−0.222794
$745$	−275850.	−0.0182088
$746$	−6.28790e6	−0.413674
$747$	2.73326e6	0.179217
$748$	−5.03539e6	−0.329063
$749$	4.92134e6	0.320538
$750$	562500.	0.0365148
$751$	−8.25040e6	−0.533796	−0.266898	−	0.963725i	$-0.585999\pi$
$751$	−8.25040e6	−0.533796	−0.266898	+	0.963725i	$0.585999\pi$
$752$	5.60333e6	0.361328
$753$	−1.30536e7	−0.838965
$754$	1.98946e6	0.127440
$755$	−1.13192e7	−0.722684
$756$	−373248.	−0.0237516
$757$	5.05745e6	0.320769	0.160384	−	0.987055i	$-0.448727\pi$
$757$	5.05745e6	0.320769	0.160384	+	0.987055i	$0.448727\pi$
$758$	1.96813e6	0.124417
$759$	1.77109e6	0.111593
$760$	−1.97120e6	−0.123793
$761$	1.26966e7	0.794742	0.397371	−	0.917658i	$-0.369923\pi$
$761$	1.26966e7	0.794742	0.397371	+	0.917658i	$0.369923\pi$
$762$	−3.93278e6	−0.245365
$763$	873280.	0.0543053
$764$	1.05504e7	0.653936
$765$	1.71315e6	0.105838
$766$	−7.31030e6	−0.450156
$767$	−728856.	−0.0447356
$768$	−589824.	−0.0360844
$769$	2.14955e7	1.31079	0.655393	−	0.755288i	$-0.272503\pi$
$769$	2.14955e7	1.31079	0.655393	+	0.755288i	$0.272503\pi$
$770$	−1.19040e6	−0.0723546
$771$	−6.49096e6	−0.393254
$772$	−1.05039e7	−0.634319
$773$	2.25149e7	1.35525	0.677626	−	0.735406i	$-0.263009\pi$
$773$	2.25149e7	1.35525	0.677626	+	0.735406i	$0.263009\pi$
$774$	4.05778e6	0.243465
$775$	3.65000e6	0.218292
$776$	8.01114e6	0.477573
$777$	1.11226e6	0.0660925
$778$	−1.57924e7	−0.935404
$779$	−1.94336e7	−1.14738
$780$	−1.37520e6	−0.0809337
$781$	8.40571e6	0.493114
$782$	1.79014e6	0.104681
$783$	949158.	0.0553266
$784$	−4.04045e6	−0.234768
$785$	−9.28685e6	−0.537891
$786$	1.91160e6	0.110367
$787$	3.26608e7	1.87971	0.939853	−	0.341580i	$-0.110962\pi$
$787$	3.26608e7	1.87971	0.939853	+	0.341580i	$0.110962\pi$
$788$	−1.36438e7	−0.782745
$789$	−6.86059e6	−0.392346
$790$	6.63520e6	0.378256
$791$	4.48109e6	0.254649
$792$	1.92845e6	0.109243
$793$	1.32470e7	0.748057
$794$	−2.98252e6	−0.167893
$795$	−390150.	−0.0218934
$796$	6.59994e6	0.369196
$797$	−1.25271e7	−0.698561	−0.349281	−	0.937018i	$-0.613574\pi$
$797$	−1.25271e7	−0.698561	−0.349281	+	0.937018i	$0.613574\pi$
$798$	−1.41926e6	−0.0788962
$799$	−1.85172e7	−1.02615
$800$	640000.	0.0353553
$801$	2.96897e6	0.163503
$802$	2.03495e7	1.11717
$803$	2.14279e7	1.17271
$804$	−3.29645e6	−0.179848
$805$	423200.	0.0230174
$806$	−8.92352e6	−0.483836
$807$	1.45068e7	0.784131
$808$	5.52922e6	0.297944
$809$	1.86887e7	1.00394	0.501971	−	0.864884i	$-0.332608\pi$
$809$	1.86887e7	1.00394	0.501971	+	0.864884i	$0.332608\pi$
$810$	−656100.	−0.0351364
$811$	4.14348e6	0.221214	0.110607	−	0.993864i	$-0.464721\pi$
$811$	4.14348e6	0.221214	0.110607	+	0.993864i	$0.464721\pi$
$812$	−666624.	−0.0354806
$813$	7.35343e6	0.390179
$814$	−5.74666e6	−0.303986
$815$	8.94190e6	0.471559
$816$	1.94918e6	0.102477
$817$	1.54296e7	0.808721
$818$	8.51377e6	0.444876
$819$	−990144.	−0.0515809
$820$	6.30960e6	0.327693
$821$	2.00599e7	1.03865	0.519327	−	0.854576i	$-0.326183\pi$
$821$	2.00599e7	1.03865	0.519327	+	0.854576i	$0.326183\pi$
$822$	1.01859e7	0.525800
$823$	1.70837e7	0.879188	0.439594	−	0.898197i	$-0.355122\pi$
$823$	1.70837e7	0.879188	0.439594	+	0.898197i	$0.355122\pi$
$824$	9.14893e6	0.469410
$825$	−2.09250e6	−0.107036
$826$	244224.	0.0124548
$827$	9.20222e6	0.467874	0.233937	−	0.972252i	$-0.424839\pi$
$827$	9.20222e6	0.467874	0.233937	+	0.972252i	$0.424839\pi$
$828$	−685584.	−0.0347524
$829$	2.73405e7	1.38172	0.690861	−	0.722988i	$-0.257232\pi$
$829$	2.73405e7	1.38172	0.690861	+	0.722988i	$0.257232\pi$
$830$	−3.37440e6	−0.170021
$831$	−1.42914e7	−0.717916
$832$	−1.56467e6	−0.0783637
$833$	1.33524e7	0.666726
$834$	−1.32346e7	−0.658864
$835$	−1.33452e7	−0.662383
$836$	7.33286e6	0.362876
$837$	−4.25736e6	−0.210052
$838$	2.17607e7	1.07044
$839$	3.02621e7	1.48421	0.742104	−	0.670285i	$-0.233828\pi$
$839$	3.02621e7	1.48421	0.742104	+	0.670285i	$0.233828\pi$
$840$	460800.	0.0225328
$841$	−1.88159e7	−0.917352
$842$	−4.92412e6	−0.239358
$843$	9.30285e6	0.450866
$844$	1.62197e7	0.783767
$845$	5.63422e6	0.271452
$846$	7.09171e6	0.340663
$847$	−725344.	−0.0347405
$848$	−443904.	−0.0211982
$849$	5.04558e6	0.240238
$850$	−2.11500e6	−0.100407
$851$	2.04300e6	0.0967039
$852$	−3.25382e6	−0.153566
$853$	2.36688e7	1.11379	0.556895	−	0.830583i	$-0.311993\pi$
$853$	2.36688e7	1.11379	0.556895	+	0.830583i	$0.311993\pi$
$854$	−4.43878e6	−0.208266
$855$	−2.49480e6	−0.116713
$856$	9.84269e6	0.459123
$857$	−1.25794e6	−0.0585071	−0.0292535	−	0.999572i	$-0.509313\pi$
$857$	−1.25794e6	−0.0585071	−0.0292535	+	0.999572i	$0.509313\pi$
$858$	5.11574e6	0.237241
$859$	3.29526e7	1.52373	0.761863	−	0.647738i	$-0.224285\pi$
$859$	3.29526e7	1.52373	0.761863	+	0.647738i	$0.224285\pi$
$860$	−5.00960e6	−0.230971
$861$	4.54291e6	0.208846
$862$	1.19222e6	0.0546499
$863$	−1.04623e7	−0.478192	−0.239096	−	0.970996i	$-0.576851\pi$
$863$	−1.04623e7	−0.478192	−0.239096	+	0.970996i	$0.576851\pi$
$864$	−746496.	−0.0340207
$865$	−1.30898e7	−0.594828
$866$	2.27650e6	0.103151
$867$	6.33727e6	0.286322
$868$	2.99008e6	0.134705
$869$	−2.46829e7	−1.10879
$870$	−1.17180e6	−0.0524874
$871$	−8.74474e6	−0.390573
$872$	1.74656e6	0.0777844
$873$	1.01391e7	0.450260
$874$	−2.60691e6	−0.115438
$875$	−500000.	−0.0220775
$876$	−8.29469e6	−0.365208
$877$	3.54954e7	1.55838	0.779190	−	0.626788i	$-0.215631\pi$
$877$	3.54954e7	1.55838	0.779190	+	0.626788i	$0.215631\pi$
$878$	−1.64254e7	−0.719083
$879$	2.17158e7	0.947991
$880$	−2.38080e6	−0.103637
$881$	−3.76131e7	−1.63267	−0.816337	−	0.577575i	$-0.803999\pi$
$881$	−3.76131e7	−1.63267	−0.816337	+	0.577575i	$0.803999\pi$
$882$	−5.11369e6	−0.221342
$883$	4.47748e7	1.93256	0.966278	−	0.257500i	$-0.0828989\pi$
$883$	4.47748e7	1.93256	0.966278	+	0.257500i	$0.0828989\pi$
$884$	5.17075e6	0.222548
$885$	429300.	0.0184248
$886$	−1.21278e7	−0.519037
$887$	621288.	0.0265145	0.0132573	−	0.999912i	$-0.495780\pi$
$887$	621288.	0.0265145	0.0132573	+	0.999912i	$0.495780\pi$
$888$	2.22451e6	0.0946678
$889$	3.49581e6	0.148352
$890$	−3.66540e6	−0.155112
$891$	2.44069e6	0.102996
$892$	−2.91462e6	−0.122651
$893$	2.69660e7	1.13159
$894$	−397224.	−0.0166223
$895$	1.11795e7	0.466514
$896$	524288.	0.0218172
$897$	−1.81870e6	−0.0754711
$898$	5.84158e6	0.241735
$899$	−7.60368e6	−0.313780
$900$	810000.	0.0333333
$901$	1.46696e6	0.0602015
$902$	−2.34717e7	−0.960569
$903$	−3.60691e6	−0.147203
$904$	8.96218e6	0.364748
$905$	−4.01885e6	−0.163110
$906$	−1.62996e7	−0.659717
$907$	7.07352e6	0.285507	0.142754	−	0.989758i	$-0.454404\pi$
$907$	7.07352e6	0.285507	0.142754	+	0.989758i	$0.454404\pi$
$908$	−1.13357e6	−0.0456282
$909$	6.99791e6	0.280905
$910$	1.22240e6	0.0489339
$911$	9.33912e6	0.372829	0.186415	−	0.982471i	$-0.440313\pi$
$911$	9.33912e6	0.372829	0.186415	+	0.982471i	$0.440313\pi$
$912$	−2.83853e6	−0.113007
$913$	1.25528e7	0.498382
$914$	2.98835e7	1.18322
$915$	−7.80255e6	−0.308094
$916$	−1.86006e6	−0.0732469
$917$	−1.69920e6	−0.0667300
$918$	2.46694e6	0.0966165
$919$	−2.76880e7	−1.08144	−0.540721	−	0.841202i	$-0.681849\pi$
$919$	−2.76880e7	−1.08144	−0.540721	+	0.841202i	$0.681849\pi$
$920$	846400.	0.0329690
$921$	−2.03156e7	−0.789189
$922$	−5.35990e6	−0.207649
$923$	−8.63167e6	−0.333496
$924$	−1.71418e6	−0.0660504
$925$	−2.41375e6	−0.0927551
$926$	1.32563e7	0.508038
$927$	1.15791e7	0.442564
$928$	−1.33325e6	−0.0508207
$929$	3.75679e7	1.42816	0.714082	−	0.700063i	$-0.246845\pi$
$929$	3.75679e7	1.42816	0.714082	+	0.700063i	$0.246845\pi$
$930$	5.25600e6	0.199273
$931$	−1.94447e7	−0.735235
$932$	2.09463e7	0.789893
$933$	1.82653e7	0.686946
$934$	9.66010e6	0.362338
$935$	7.86780e6	0.294323
$936$	−1.98029e6	−0.0738820
$937$	−1.95495e7	−0.727422	−0.363711	−	0.931512i	$-0.618490\pi$
$937$	−1.95495e7	−0.727422	−0.363711	+	0.931512i	$0.618490\pi$
$938$	2.93018e6	0.108739
$939$	−2.23762e7	−0.828176
$940$	−8.75520e6	−0.323181
$941$	−2.17384e7	−0.800300	−0.400150	−	0.916450i	$-0.631042\pi$
$941$	−2.17384e7	−0.800300	−0.400150	+	0.916450i	$0.631042\pi$
$942$	−1.33731e7	−0.491025
$943$	8.34445e6	0.305575
$944$	488448.	0.0178397
$945$	583200.	0.0212441
$946$	1.86357e7	0.677046
$947$	2.97375e7	1.07753	0.538766	−	0.842455i	$-0.318891\pi$
$947$	2.97375e7	1.07753	0.538766	+	0.842455i	$0.318891\pi$
$948$	9.55469e6	0.345299
$949$	−2.20040e7	−0.793114
$950$	3.08000e6	0.110724
$951$	−1.89059e7	−0.677869
$952$	−1.73261e6	−0.0619595
$953$	−1.85027e7	−0.659937	−0.329969	−	0.943992i	$-0.607038\pi$
$953$	−1.85027e7	−0.659937	−0.329969	+	0.943992i	$0.607038\pi$
$954$	−561816.	−0.0199859
$955$	−1.64850e7	−0.584899
$956$	−1.33795e7	−0.473474
$957$	4.35910e6	0.153857
$958$	8.53843e6	0.300583
$959$	−9.05414e6	−0.317908
$960$	921600.	0.0322749
$961$	5.47645e6	0.191289
$962$	5.90114e6	0.205588
$963$	1.24572e7	0.432866
$964$	1.02478e7	0.355173
$965$	1.64124e7	0.567352
$966$	609408.	0.0210119
$967$	−3.53024e6	−0.121405	−0.0607026	−	0.998156i	$-0.519334\pi$
$967$	−3.53024e6	−0.121405	−0.0607026	+	0.998156i	$0.519334\pi$
$968$	−1.45069e6	−0.0497606
$969$	9.38045e6	0.320933
$970$	−1.25174e7	−0.427154
$971$	3.69827e6	0.125878	0.0629391	−	0.998017i	$-0.479953\pi$
$971$	3.69827e6	0.125878	0.0629391	+	0.998017i	$0.479953\pi$
$972$	−944784.	−0.0320750
$973$	1.17641e7	0.398361
$974$	9.93445e6	0.335542
$975$	2.14875e6	0.0723893
$976$	−8.87757e6	−0.298311
$977$	−2.36072e6	−0.0791239	−0.0395620	−	0.999217i	$-0.512596\pi$
$977$	−2.36072e6	−0.0791239	−0.0395620	+	0.999217i	$0.512596\pi$
$978$	1.28763e7	0.430472
$979$	1.36353e7	0.454682
$980$	6.31320e6	0.209983
$981$	2.21049e6	0.0733358
$982$	1.12488e7	0.372245
$983$	−4.47830e7	−1.47819	−0.739094	−	0.673603i	$-0.764746\pi$
$983$	−4.47830e7	−1.47819	−0.739094	+	0.673603i	$0.764746\pi$
$984$	9.08582e6	0.299141
$985$	2.13184e7	0.700108
$986$	4.40597e6	0.144327
$987$	−6.30374e6	−0.205971
$988$	−7.52998e6	−0.245415
$989$	−6.62520e6	−0.215381
$990$	−3.01320e6	−0.0977102
$991$	−2.31755e7	−0.749626	−0.374813	−	0.927100i	$-0.622293\pi$
$991$	−2.31755e7	−0.749626	−0.374813	+	0.927100i	$0.622293\pi$
$992$	5.98016e6	0.192945
$993$	2.11957e7	0.682141
$994$	2.89229e6	0.0928486
$995$	−1.03124e7	−0.330219
$996$	−4.85914e6	−0.155207
$997$	3.20121e7	1.01994	0.509972	−	0.860191i	$-0.329656\pi$
$997$	3.20121e7	1.01994	0.509972	+	0.860191i	$0.329656\pi$
$998$	3.03640e7	0.965012
$999$	2.81540e6	0.0892537

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.6.a.a.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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